Convergence

Integral Test

If f is a continuous function, it converges if and only if its integral also converges

P-series Test

\(\sum_{n=1}^\infty n^{-p}\) converges for all p > 1

Comparison Test

If a(n) is convergent and is always bigger than b(n) in an appropriate range, b(n) is also convergent.If a(n) is divergent and is always smaller than b(n) in an appropriate range, b(n) is also divergent

Limit Comparison Test

For a(n) and b(n), where i and j are their respective limits towards infinity, if i/j = c > 0 and is finite, then both functions converge or diverge (same behaviour).

Alternating Series Test

An alternating series is one where the terms switch signs for every adjacent term. The series converges if its sequence in absolute values is decreasing and if it approaches 0 as \(n \rightarrow \infty\)

\( \text{Find all values of x, y, z, and $\lambda$ such that $\nabla f(x, y, z) = \lambda \nabla g(x, y, z)$ and } g(x, y, z) = k \\ \text{Evaluate $f$ at all points (x, y, z) from the values above; the largest is the maximum and the smallest is the minimum} \)